Extended Mittag--Leffler Stability Results for Fuzzy Fractional Systems

Abstract

We extend Lyapunov--type Mittag--Leffler stability analysis for fuzzy nonlinear fractional differential equations (Caputo sense) and introduce a family of stronger, broadly applicable stability results. In particular, we develop (i) uniform Mittag--Leffler stability for variable-order Caputo derivatives, (ii) input-to-state stability (ISS) in the Mittag--Leffler sense and explicit robustness/ultimate bound estimates, (iii) a converse Mittag--Leffler Lyapunov theorem, (iv) a fractional LaSalle invariance principle adapted to the fuzzy setting, and (v) practical and computational criteria including Lyapunov--Krasovskii functionals for delay systems and levelwise LMI tests for linearized models. The paper presents precise assumptions, proof sketches, and preparatory lemmas in the preliminaries; subsequent sections (Step 2 onward) provide rigorous proofs and illustrative examples.